This article illustrates how to calculate percentiles manually and step-by-step, a useful statistical measure for understanding your data. No software or coding skills are needed: you only need a piece of paper, a pen, a calculator, and a willingness to do some simple math!
What are Percentiles?
Percentiles are a central tool in statistics. They help understand how data points compare within a dataset in terms of value ranges. Closely related to the notion of percentage, a percentile tells us the value in a data set below which a determined percentage of instances fall. For example, the 50th percentile in a dataset of numerical exam results for a student cohort tells us the “threshold” mark below which 50% of students in the cohort fall. Concretely, if this 50th percentile were 65 over 100, that would mean half of the students have marks below 65. Easy, isn’t it?
Unlike averages or medians, which give us indicators of the “center” of the data differently interpreted, percentiles go one step beyond, allowing us to identify where specific points lie concerning specific portions of the data. In fact, the median is nothing other than the 50th percentile — see example above — but percentiles give access to more detailed and specific positions of data points across its entire range of values.
Besides the median, two more frequently used types of percentiles are:
Deciles: they divide the data set into ten equal parts, each representing 10% buckets of the data. This plot illustrates the location of the ten deciles in a dataset via its histogram: percentiles are depicted by vertical dashed lines.
Quartiles: they divide the data set into four equal parts, each representing 25% buckets of the data. The box plot is a particular type of visualization approach to display quartiles (blue dashed lines indicate where the 1st, 2nd, and 3rd quartiles lie):
Step-by-Step Percentile Calculation
Let’s follow these four simple steps to calculate any percentiles on your data. The example assumes having single-variable data points. These could be, for instance, commuting times from home to work, collected from a group of employees living in the same district of a large city.
- Organize the data: first, the data points in the sample must be arranged in ascending order. Suppose we collected commuting times (in minutes) from nine employees, then the arranged data values could look like this: 55, 63, 70, 72, 80, 85, 90, 92, and 95.
- Given the percentile, find the rank position: Let’s suppose we want to find the 75th percentile of the data, in other words, the travel duration below which 75% of employees manage to get to the office. The rank (which we will explain a bit later) is computed using the following formula, where P is the percentile specified and N is the number of data points. In our example, P=75, and N=9:
R = (P / 100) × (N + 1) = (75 / 100) × (9 + 1) = 7.5
- Interpret the rank: So, what does this R=7.5 tell us? The rank indicates the position in the sorted dataset (see step 1) of the percentile value we are looking for. If R is an integer value, we just locate that position in the sorted data. Meanwhile, if R is not an integer, e.g. R=7.5, we must locate the two consecutive values with integer positions above and below R, in this case, the 7th and 8th values, and average them to find the percentile we are looking for:
55, 63, 70, 72, 80, 85, 90, 92, 95
Since the 7th and 8th positions correspond with the values 90 and 92, the 75th percentile sought is (90 + 92)/2 = 91.
- Interpret the results: the 75th percentile score obtained in this example is 91, which means that 75% of commuters’ time to the office is below 91 minutes.
Now that you learned how to calculate percentiles by hand, try calculating by hand the 40th and 80th percentiles of this dataset: you should get 72 and 92 respectively.
For a final remark, if you also try getting the 95th percentile, you’ll come across R=9.5, meaning a rank position between the 9th and the 10th sorted data point. The problem? We only have N=9 data points! In these situations, we simply assume that the extra 10th position would contain the same value as the one in the last position again, i.e. 95. Hence the resulting percentile would be 95.
